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Properties

Label 16-91e16-1.1-c1e8-0-3
Degree $16$
Conductor $2.211\times 10^{31}$
Sign $1$
Analytic cond. $3.65493\times 10^{14}$
Root an. cond. $8.13167$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $8$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 6·4-s − 4·8-s − 10·9-s − 4·11-s − 35·16-s − 40·18-s − 16·22-s − 12·23-s − 26·25-s − 8·29-s − 60·32-s − 60·36-s − 8·37-s − 32·43-s − 24·44-s − 48·46-s − 104·50-s − 4·53-s − 32·58-s − 14·64-s + 20·67-s + 8·71-s + 40·72-s − 32·74-s − 4·79-s + 33·81-s + ⋯
L(s)  = 1  + 2.82·2-s + 3·4-s − 1.41·8-s − 3.33·9-s − 1.20·11-s − 8.75·16-s − 9.42·18-s − 3.41·22-s − 2.50·23-s − 5.19·25-s − 1.48·29-s − 10.6·32-s − 10·36-s − 1.31·37-s − 4.87·43-s − 3.61·44-s − 7.07·46-s − 14.7·50-s − 0.549·53-s − 4.20·58-s − 7/4·64-s + 2.44·67-s + 0.949·71-s + 4.71·72-s − 3.71·74-s − 0.450·79-s + 11/3·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(3.65493\times 10^{14}\)
Root analytic conductor: \(8.13167\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(8\)
Selberg data: \((16,\ 7^{16} \cdot 13^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( ( 1 - p T + 3 T^{2} - T^{4} + 3 p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} )^{2} \)
3 \( 1 + 10 T^{2} + 67 T^{4} + 34 p^{2} T^{6} + 1057 T^{8} + 34 p^{4} T^{10} + 67 p^{4} T^{12} + 10 p^{6} T^{14} + p^{8} T^{16} \)
5 \( 1 + 26 T^{2} + 311 T^{4} + 2358 T^{6} + 13281 T^{8} + 2358 p^{2} T^{10} + 311 p^{4} T^{12} + 26 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + 2 T + 39 T^{2} + 54 T^{3} + 611 T^{4} + 54 p T^{5} + 39 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
17 \( 1 + 78 T^{2} + 3142 T^{4} + 85520 T^{6} + 1694711 T^{8} + 85520 p^{2} T^{10} + 3142 p^{4} T^{12} + 78 p^{6} T^{14} + p^{8} T^{16} \)
19 \( 1 + 58 T^{2} + 1703 T^{4} + 44478 T^{6} + 996825 T^{8} + 44478 p^{2} T^{10} + 1703 p^{4} T^{12} + 58 p^{6} T^{14} + p^{8} T^{16} \)
23 \( ( 1 + 6 T + 78 T^{2} + 358 T^{3} + 2630 T^{4} + 358 p T^{5} + 78 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 + 4 T + 31 T^{2} + 158 T^{3} + 1897 T^{4} + 158 p T^{5} + 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 + 178 T^{2} + 15406 T^{4} + 835752 T^{6} + 30991415 T^{8} + 835752 p^{2} T^{10} + 15406 p^{4} T^{12} + 178 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 + 4 T + 62 T^{2} + 48 T^{3} + 1470 T^{4} + 48 p T^{5} + 62 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 + 152 T^{2} + 6780 T^{4} - 92120 T^{6} - 15043642 T^{8} - 92120 p^{2} T^{10} + 6780 p^{4} T^{12} + 152 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 + 16 T + 157 T^{2} + 1020 T^{3} + 6599 T^{4} + 1020 p T^{5} + 157 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 + 150 T^{2} + 11494 T^{4} + 707936 T^{6} + 37180919 T^{8} + 707936 p^{2} T^{10} + 11494 p^{4} T^{12} + 150 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 2 T + 132 T^{2} + 248 T^{3} + 8645 T^{4} + 248 p T^{5} + 132 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 + 278 T^{2} + 36286 T^{4} + 3136752 T^{6} + 207469775 T^{8} + 3136752 p^{2} T^{10} + 36286 p^{4} T^{12} + 278 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 + 380 T^{2} + 67856 T^{4} + 7443748 T^{6} + 547986382 T^{8} + 7443748 p^{2} T^{10} + 67856 p^{4} T^{12} + 380 p^{6} T^{14} + p^{8} T^{16} \)
67 \( ( 1 - 10 T + 262 T^{2} - 1788 T^{3} + 25847 T^{4} - 1788 p T^{5} + 262 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
71 \( ( 1 - 4 T + 118 T^{2} - 584 T^{3} + 12630 T^{4} - 584 p T^{5} + 118 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 + 156 T^{2} + 7232 T^{4} + 2020 p T^{6} + 11253390 T^{8} + 2020 p^{3} T^{10} + 7232 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 + 2 T + 38 T^{2} - 238 T^{3} + 1686 T^{4} - 238 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 + 314 T^{2} + 46430 T^{4} + 4837464 T^{6} + 427877847 T^{8} + 4837464 p^{2} T^{10} + 46430 p^{4} T^{12} + 314 p^{6} T^{14} + p^{8} T^{16} \)
89 \( 1 + 106 T^{2} + 17871 T^{4} + 2077238 T^{6} + 155830553 T^{8} + 2077238 p^{2} T^{10} + 17871 p^{4} T^{12} + 106 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 + 412 T^{2} + 82235 T^{4} + 11265530 T^{6} + 1213308829 T^{8} + 11265530 p^{2} T^{10} + 82235 p^{4} T^{12} + 412 p^{6} T^{14} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−3.57191815387198412914385069444, −3.48675095998826300482386618127, −3.33372872168128179600693721315, −3.29203663338605281611798135820, −3.25900175160945153815248509171, −3.25694947920206463166281019181, −3.12594273772106451824237466022, −2.80714235144339393966805671049, −2.67370695667621034556154409874, −2.64411657839075160415144596788, −2.56951599642367372914163629096, −2.51373308112053886567142665535, −2.37501971236435921930906531474, −2.26847660556988742926604096231, −2.22442549131450252568618400870, −1.98896172736970333578485648921, −1.96670978931651905700950370637, −1.88995924545262765776776341711, −1.74497831279056286756936142117, −1.68702504986466060437456444327, −1.50693270523256377336847985865, −1.09549482425138947861276967037, −1.03453572755467412603791511356, −0.955618115093120839516226605488, −0.819433988137413312919178915922, 0, 0, 0, 0, 0, 0, 0, 0, 0.819433988137413312919178915922, 0.955618115093120839516226605488, 1.03453572755467412603791511356, 1.09549482425138947861276967037, 1.50693270523256377336847985865, 1.68702504986466060437456444327, 1.74497831279056286756936142117, 1.88995924545262765776776341711, 1.96670978931651905700950370637, 1.98896172736970333578485648921, 2.22442549131450252568618400870, 2.26847660556988742926604096231, 2.37501971236435921930906531474, 2.51373308112053886567142665535, 2.56951599642367372914163629096, 2.64411657839075160415144596788, 2.67370695667621034556154409874, 2.80714235144339393966805671049, 3.12594273772106451824237466022, 3.25694947920206463166281019181, 3.25900175160945153815248509171, 3.29203663338605281611798135820, 3.33372872168128179600693721315, 3.48675095998826300482386618127, 3.57191815387198412914385069444

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.